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Differential Equations

Short Answer Type

311. $If space straight y space equals space Pe to the power of ax plus Qe to the power of bx$ show that
$straight d to the power of 2 straight y end exponent over dx squared minus left parenthesis straight a plus straight b right parenthesis dy over dx plus aby space equals 0$

Given that

$straight y equals Pe to the power of ax plus Qe to the power of bx Differentiating space the space above space function space with space respect space to space straight x comma dy over dx equals Pae to the power of ax plus Qbe to the power of bx Differentiating space once space again comma space we space have comma fraction numerator straight d squared straight y over denominator dx squared end fraction equals space Pa squared straight e to the power of ax plus Qb squared straight e to the power of bx$
Let us now find $left parenthesis straight a plus straight b right parenthesis dy over dx colon$

$left parenthesis straight a plus straight b right parenthesis dy over dx equals left parenthesis straight a plus straight b right parenthesis open parentheses Pae to the power of ax plus Qbe to the power of bx close parentheses space space rightwards double arrow space left parenthesis straight a plus straight b right parenthesis dy over dx equals Pa squared straight e to the power of ax plus Qabe to the power of bx plus Pabe to the power of bx plus Qb squared straight e to the power of bx rightwards double arrow space left parenthesis straight a plus straight b right parenthesis dy over dx equals Pa squared straight e to the power of ax plus left parenthesis straight P plus straight Q right parenthesis abe to the power of bx plus Qb squared straight e to the power of bx space space space$
Also we have,

$aby space equals ab open parentheses Pe to the power of ax plus Qe to the power of bx close parentheses space equals space ab Thus comma space fraction numerator straight d squared straight y over denominator dx squared end fraction minus left parenthesis straight a plus straight b right parenthesis dy over dx plus aby equals Pa squared straight e to the power of ax plus Qb squared straight e to the power of bx minus Pa squared straight e to the power of ax minus left parenthesis straight P plus straight Q right parenthesis abe to the power of bx minus Qb squared straight e to the power of bx plus abPe to the power of ax plus abQe to the power of bx equals 0$

111 Views

312.

If space straight x equals cost left parenthesis 3 minus 2 cos squared straight t right parenthesis space and space straight y equals sint left parenthesis 3 minus 2 sin squared straight t right parenthesis comma space find space the space value space of space dy over dx at space straight t equals straight pi over 4.

107 Views

313.

Find the particular solution of the differential equation $log open parentheses dy over dx close parentheses equals 3 straight x plus 4 straight y comma space given space that space straight y space equals space 0 comma space when space straight x equals space 0.$

120 Views

314.

Write the degree of the differential equation $straight x cubed open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses squared space plus space open parentheses dy over dx close parentheses to the power of 4 space equals space 0$

180 Views

315.

The amount of pollution content added in air in city due to x-diesel vehicles is given by $straight P left parenthesis straight x right parenthesis space equals space 0.005 straight x cubed plus 0.02 straight x squared plus 30 straight x.$ Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions.

225 Views

316.

Show that the function f in $straight A space equals space straight R to the power of minus open curly brackets 2 over 3 close curly brackets space defined space as space straight f left parenthesis straight x right parenthesis space equals space fraction numerator 4 straight x plus 3 over denominator 6 straight x minus 4 end fraction space is space one minus one space and space onto. space Hence space find space straight f to the power of negative 1 end exponent.$

130 Views

317.

Differentiate the following function with respect to x:
$left parenthesis log space straight x right parenthesis to the power of straight x plus straight x to the power of log straight x$

142 Views

318.

If space straight y space equals space log space open square brackets straight x space plus space square root of straight x squared plus space straight a squared end root close square brackets comma space show space that space left parenthesis straight x squared space plus space straight a squared right parenthesis space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus straight x dy over dx space equals space 0

186 Views

319.

If $straight x equals straight a space sint space and space straight y space equals space straight a open parentheses cost plus log space tan straight t over 2 close parentheses space find space fraction numerator straight d squared straight y over denominator dx squared end fraction$

500 Views

Long Answer Type

320.

Show that the differential equation $2 ye to the power of straight x divided by straight y end exponent dx plus left parenthesis straight y minus 2 straight x space straight e to the power of straight x divided by straight y end exponent right parenthesis space dy space equals space 0$ is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.